Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13313 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#13314 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | #466 |
| commutator, Frattini, cosocle | C22×C8 | C23×C5 |
| prime radical | C2×C5×#472 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C22×C4×C5 ◁ C2×C5×#131 ◁ G | |
536 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 3 generators on 41 points:
(1,2,3,4,5)(6,7)(8,9)(10,11,16,22)(12,23,27,36)(13,17,30,33)(14,21,15,20)(18,24,37,26)(19,29,32,28)(25,35,40,34)(31,39,38,41);(11,17)(12,24)(13,28)(14,15)(18,34)(19,20)(21,32)(22,33)(23,38)(25,40)(26,27)(29,30)(31,36)(35,37);(7,9)(10,12)(11,18)(13,25)(14,24)(15,26)(16,27)(17,31)(19,35)(20,36)(21,23)(22,37)(28,39)(29,41)(30,40)(32,34)(33,38)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 6302);